model, calibration, and paramater risk · 2014-01-16 · moment matching calibration •using the...
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MODEL,
CALIBRATION, AND
PARAMATER RISK
Wim SCHOUTENS
Aarhus Quant Factory – Symposium
Aarhus Quant Day
17 January, 2014
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CONTENT
• What is Calibration ?
• Model Risk : A Perfect Calibration ! Now What ?
• Calibration Risk
• New Calibration Methods : Moment Matching
• Conic Finance Perspective
• Joint work with: Florence Guillaume, Jurgen Tistaert, Erwin Simons, Dilip Madan.
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What is Calibration ?
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What is Calibration ?
• Calibration in a nutshell :
Derivatives
market prices
Derivatives
model pricesFind:
model parameters that
match
model prices
as best as possible with
market prices
Optimization problem:
Finding minimum “distance”
Pricing model Model parameters
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What is Calibration ?
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Calibration in Equity Land
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Calibration in E2C land
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Calibration in CDO Land
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Model and Calibration Risk
• Model Risk
One can calibrate different advanced models to a given
market setting.
Suppose the calibration fits are good.
One can then price exotics under the calibrated model.
Different models can give rise to significant different exotic
prices.
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Model and Calibration Risk
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Model and Calibration Risk
• Calibration Risk
In the calibration procedure one has to make different
choices:• Which function to minimize (rmse, ape, aae, arpe, …) ?
• Which vanilla to calibrated on (ATM, OTM, …) ?
• Use a weighting and if so how to weight ?
• Calibrate on prices or on implied volatilities ?
• Which starting value to take (yesterday’s optimum, random, historical,…)?
• Which search algorithm to use (Nelder-Nead, Gauss, Newton Gradient
decent,….) ?
• Pre-fix some parameters and how ?
Suppose the calibration under different choices are good,
but lead to different optimal parameters.
Different parameter settings can give rise to significant
different exotic prices.
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Model and Calibration Risk
• Example: The Heston Model
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Model and Calibration Risk• Calibration Risk: The choice of the objective function
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Model and Calibration Risk
• Calibration Risk: pre-fixing
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Model and Calibration Risk
• Calibration Risk
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Model and Calibration Risk
• Calibration Risk
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Model and Calibration Risk
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Model and Calibration Risk
• Calibration Risk
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Model and Calibration Risk
• Calibration Risk
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Moment Matching Calibration
• Using the implied moment formula (cfr. VIX calculation)
• One can back out of the option price surface the implied
variance, skewness, kurtosis and higher moments of the
risk-neutral return distribution.
• Since the above moments are also known in close form for
many popular models (Heston, VG, …), one tries to fit
these moments as best as possible.
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Moment Matching Calibration
• Moment matching Calibration
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Moment Matching Calibration
• For VG, we have
• Knowing the statistics and solving for the parameters:
where v, s and k denotes the market implied variance,
skewness and kurtosis of the log asset return for the time
horizon T.
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Moment Matching Calibration
• The system admits a solution iff
• If no perfect matching solution exists one can look for the
“closest” one in the existence domain.
• MAIN ADVANTAGES
o No integration is needed
o Extremely fast
o No initial parameter needed
o No search-algorithm involved
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Moment Matching Calibration
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Moment Matching Calibration
• VG-Term structure model average results
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Moment Matching Calibration
• Heston – VS calibration
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Conic Finance
• Bid-Ask Pricing
o We will make use of the minmaxvar distortion function:
o We use non linear expectation to calculate (bid and ask) prices.
o The distorted expectation of a random variable with distribution
function F(x) is defined
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Conic Finance
• Bid-Ask Pricing
The ask price of payoff X is determined as
The bid price of payoff X is determined as
Hence for the BID price we have put more weight on the
down-side. For the ASK the upside has been receiving
more weighted.
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Conic Finance
MC:
DISTORTED UNIFORM (1/N) WEIGHTS:
Equally weighted
Distorted weights
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Conic Finance
• Bid-Ask Pricing
These formulas are derived by noting that the cash-flow of selling at its
ask price and buying at its bid price is acceptable in the relevant
market .
We say that a risk/cash-flow X is acceptable if
M is a set of test-measures under which cash-flows need to have
positive expectation.
Operational cones were defined by Cherney and Madan and depend
solely on the distribution function G(x) of X and a distortion function ϕ.
One can show that we need to have that the distorted expectation is
positive:
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Conic Finance
• How does everything relate to each other ?
Model Risk : different models – different Q measures – different exotics
Calibration Risk : same model – different parameters/Q – different exotics.
Buying X at its bid price is acceptable :
Selling X at its ask price is acceptable :
Ask price is supremum of test-measure prices :
Bid price is infimum of test-measure prices:
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Conclusion
• Calibration is an important procedure in derivatives pricing.
• Model risk and calibration risks are omnipresent and can lead to wide
ranges in exotic pricing.
• Alternative calibration methods are currently developed.
• Moment matching calibration is not using a search algorithm and
hence also not a starting value. It relies on closed-form expressions of the
underlying moments, which can be readily estimated from market quotes.
It is extremely fast.
• Conic finance puts model and calibration risk into a bid-ask
perspective. The more uncertainty there is on the model/calibration to be
used the wider the bid-ask spread.
More info: www.schoutens.be
Email: [email protected]
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Thank you for your
attention !